521 research outputs found
Global convergence rate analysis of unconstrained optimization methods based on probabilistic models
We present global convergence rates for a line-search method which is based
on random first-order models and directions whose quality is ensured only with
certain probability. We show that in terms of the order of the accuracy, the
evaluation complexity of such a method is the same as its counterparts that use
deterministic accurate models; the use of probabilistic models only increases
the complexity by a constant, which depends on the probability of the models
being good. We particularize and improve these results in the convex and
strongly convex case.
We also analyze a probabilistic cubic regularization variant that allows
approximate probabilistic second-order models and show improved complexity
bounds compared to probabilistic first-order methods; again, as a function of
the accuracy, the probabilistic cubic regularization bounds are of the same
(optimal) order as for the deterministic case
A Generic Approach for Escaping Saddle points
A central challenge to using first-order methods for optimizing nonconvex
problems is the presence of saddle points. First-order methods often get stuck
at saddle points, greatly deteriorating their performance. Typically, to escape
from saddles one has to use second-order methods. However, most works on
second-order methods rely extensively on expensive Hessian-based computations,
making them impractical in large-scale settings. To tackle this challenge, we
introduce a generic framework that minimizes Hessian based computations while
at the same time provably converging to second-order critical points. Our
framework carefully alternates between a first-order and a second-order
subroutine, using the latter only close to saddle points, and yields
convergence results competitive to the state-of-the-art. Empirical results
suggest that our strategy also enjoys a good practical performance
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